Neighborhood of a Point

neighborhood of a point

[′nā·bər‚hu̇d əv ə ′pȯint]
(mathematics)
A set in a topological space which contains an open set which contains the point; in Euclidean space, an example of a neighborhood of a point is an open (without boundary) ball centered at that point.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Neighborhood of a Point

 

(for a metric space), the set of all points whose distance from a given point is less than some positive number R. A neighborhood of this type is called spherical, and the number R is called the radius of the neighborhood. Also frequently considered are rectangular neighborhoods in a plane and their analogs in spaces of any number of dimensions. Sometimes the neighborhood of a point on a line is understood to mean any interval that includes this point, and the neighborhood of a point on a plane is understood to mean any open circle that includes this point but, perhaps, does not have the point as a center. These and other special types of neighborhoods are special cases of more general neighborhoods that are understood to mean all open sets that contain the given point.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
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